5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.
- Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
- Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.